Fully relativistic lattice Boltzmann algorithm

Abstract

Starting from the Maxwell-Jüttner equilibrium distribution, we develop a relativistic lattice Boltzmann (LB) algorithm capable of handling ultrarelativistic systems with flat, but expanding, spacetimes. The algorithm is validated through simulations of quark-gluon plasma, yielding excellent agreement with hydrodynamic simulations. The present scheme opens the possibility of transferring the recognized computational advantages of lattice kinetic theory to the context of both weakly and ultra-relativistic systems.PACS numbers: 47.75.+f, 47.11.-j Keywords: Relativistic fluid dynamics, Quark-gluon plasmas, Lattice Boltzmann I. MOTIVATIONThe great success of the Relativistic Heavy-Ion Collider (RHIC) experimental program [1][2][3][4] has provided the motivation to come up with realistic and quantitative simulations of heavy-ion collisions.Since the bulk of particles produced in relativistic heavy-ion collisions is described by fluid dynamics [5], the center-piece of any complete simulation attempt will involve a viscous fluid dynamics algorithm. The majority of presently available fluid dynamics codes is able to handle smooth initial conditions in 2+1 dimensions in the presence of shear viscosity [6][7][8][9][10]. However, it has by now been understood that the presence of event-by-event fluctuations in the initial state can lead to significantly different quantitative results with respect to smooth initial conditions [11], and may in some cases even explain qualitatively new phenomena. To be more specific, the presence of event-by-event fluctuations is the source of the non-vanishing elliptic flow found in RHIC experiments at central collisions, the source of hydrodynamic flow-fluctuations, and may (through the presence of socalled triangular flow v 3 ) naturally explain the presence of the 'ridge phenomenon' found in experiments [12][13][14][15]. Thus, it is probably fair to say that without including the effect of event-by-event fluctuations, a description of the medium created in heavy-ion collisions cannot be regarded as realistic. This provides the motivation to develop a fully relativistic and computationally efficient viscous fluid dynamics algorithm that can handle initial state fluctuations. Also, such an algorithm can be used Further motivation is provided by other systems where relativistic viscous fluid flows are of interest, such as astrophysical systems and condensed matter systems such as graphene [17]. One particular question that arises in all this different systems is when relativistic fluid flow becomes turbulent, which involves a determination of the critical Reynolds number and the turbulent spectrum [18,19]. II. LATTICE KINETIC APPROACH TO HYDRODYNAMICSFluid turbulence, both classical and relativistic, sets one of the most compelling challenges in modern computational physics. This motivates a relentless search for new and ever more efficient methods for solving the hydrodynamic equations of motion in the high-Reynolds turbulent regimes. In the last two decades, a new computational paradigm has...

Relativistic lattice Boltzmann model with improved dissipation

“…We first expand the equilibrium distribution at rest, w(p 0 ) = f eq = A exp(−p 0 ) by using Cartesian coordinates, unlike the spherical coordinate system used in Ref. [13], and choose the 4-momentum vectors such that they belong to the lattice (from now on and without loss of generality, we will use the notation p 0 /T → p 0 ). This procedure also avoids extra terms in the product, P i (θ l )R j (p l )F k (φ l ) for the spherical case, which are not necessary if we only need to recover correctly the first three moments of the equilibrium distribution.…”

“…Following Romatschke [13], we can expand the distribution in each coordinate separately, and subsequently, by using a Gauss quadrature, calculate the discrete values of the 4-momentum vectors. Thus, the discrete equilibrium dis-tribution can be written as,…”

“…According to a standard procedure [13,15,16], we first expand the Maxwell-Jüttner distribution in the rest frame, f eq = A exp(−p 0 /T ), in an orthogonal basis. Since in the ultrarelativistic regime, p 0 /T = p 2 /T 2 + m 2 /T 2 ≃ p/T , being p = p 2 , we can write the equilibrium distribution in spherical coordinates,…”

“…In order to set up a theoretical background for the lattice version of the relativistic Boltzmann equation, Romatschke et al [13] developed a scheme for an ultrarelativistic gas based on the expansion on orthogonal polynomials of the Maxwell-Jüttner distribution [14] and, by following a Gauss-type quadrature procedure, the discrete version of the distribution and the weighting func-tions was calculated. This procedure was similar to the one used for the non-relativistic lattice Boltzmann model [15,16].…”

Onset of cavitation in the quark-gluon plasma

“…Ultimately, and maybe most importantly, it would be interesting to calculate the particle spectra from a numerical simulation including the presence of hadron gas bubbles. This could potentially be done using stateof-the-art numerical hydrodynamical solvers [5,32,33] and could verify the assumption that cavitating fluids are inconsistent with experimental data on heavy-ion collisions.…”